Since the Kobayashi metric is always invariant under the group of biholomorphic automorphisms, see [1]} or [2]}, the complex orbifold \(\mathcal {M}_g\)  is also Kobayashi hyperbolic. Let \(V\)  be a hyperbolic Riemann surface, i.e. \(V \cong \mathbb {H}/G\)  where \(G\)  is a discrete subgroup of \(\mathbb {P}\mathrm {SL}(2, \mathbb {R})\) , and let
\(\phi \colon V \rightarrow \mathcal {M}_g\) 
